ON THE QUESTION OF THE FUNDAMENTAL LAW OF ADIABATIC PHASE FORMATION

S. V. Polozhii

(Presented by the scientific seminar of the Department of Thermal Power Engineering)

As is known, adiabatic phase formation is understood as a change in the state of aggregation of a heated liquid, saturated vapor, or two-phase liquid–vapor system without heat exchange with the external environment, by means of a pressure drop in the initial system.

Adiabatic phase formation is a generalized process and, in terms of its final phase-forming effect, may proceed in three directions, depending on the initial state of the system:

with an isoentropic pressure drop of a heated liquid or a two-phase system with predominance of the liquid phase, the vapor phase is formed at the end of the process—adiabatic vapor formation (AVF);

with an isoentropic pressure drop of saturated vapor or a two-phase system with predominance of the vapor phase, a liquid phase is formed—adiabatic condensation (AC);

with an isoentropic pressure drop of a two-phase system of equal weight ratio \((X = У = 0.5)\), the process proceeds without a visible change in the state of aggregation, both in intermediate and in final states—the phase ratio does not change for any magnitude of the pressure drop, \(X = У = 0.5 = \mathrm{const}\)—a process of equal phase formation.

The basic properties of adiabatic phase formation in all its modifications are: formation of a new phase, kinetic energy, and dispersion of the flow.

The formation of a new resultant phase at the end of the process under an isoentropic pressure drop—of saturated vapor or liquid—always occurs in the direction opposite to the initial phase in a single-phase initial state, or to the predominant phase in the initial two-phase state of the system.

The kinetic energy of the flow in adiabatic phase formation may have different magnitude and depends on the corresponding organization of the process, on the initial state of the phase before the phase-forming element—saturated vapor, saturated liquid, two-phase state and the ratio between the phases—as well as on the magnitude and interval of the pressure drop, and on the initial and final pressure.

The dispersion of the adiabatic phase-formation flow also depends on many factors and increases with an increase in the pressure interval, the magnitude of the pressure drop, and with an increase in the initial pressure, and also depends on the initial state of the liquid–vapor system, the two-phase system, and a number of other factors.

All these features and properties of adiabatic phase formation are consequences of the fundamental law of this process of nature.

During an isoentropic pressure drop of a single-phase or two-phase H₂O system, two opposite processes occur simultaneously—adiabatic condensation and adiabatic vaporization, with the same degree of intensity and in equal quantities per unit weight of the liquid or vapor phase, with the formation of a new phase, kinetic energy, and dispersion of the two-phase flow.

A natural source of adiabatic phase formation is the highly compressed heated waters, steam, or a two-phase system that are widespread in nature. In regions of active volcanoes and in seismic regions, the process of adiabatic phase formation occurs under natural conditions in the form of geysers or steam “fountains,” with the manifestation of the basic properties of this process—the generation of a new phase, the kinetic energy of the flow, and dispersion of the two-phase flow. When the pressure of heated, highly compressed water is released, adiabatic vaporization takes place; when a two-phase system or saturated steam exits through a fissure or borehole, the process of adiabatic condensation takes place. The quantity of the new phase, the kinetic energy of the flow, and its dispersity also depend on the magnitude and character of the pressure drop, on the initial state of the system, and on hydraulic losses, which to one degree or another may differ from an isoentropic pressure drop and affect the ratio of the phase-forming and energetic effects of this process.

As experimental investigations of adiabatic phase formation, experimental studies, and the practice of using kinetic energy in wet-steam turbines show, a high degree of completion of the adiabatic phase-formation process is obtained, and the kinetic energy of the process in magnitude proves to be close to the theoretical value for an isoentropic process. These circumstances made it possible to use thermodynamic dependences and to substantiate theoretically the fundamental law of adiabatic phase formation [1, 2, 3]. With an isoentropic pressure drop of one kg of saturated steam, the quantity of liquid at the end of the completed process is determined from the relation

\[ Y = \frac{S_0'' - S_1'}{r_1} T_{\mathrm{n}1}. \]

With an isoentropic pressure drop of kg of saturated liquid, at the end of the completed process steam is formed,

\[ X_1 = \frac{S_0' - S_1'}{r_1} T_{\mathrm{n}1}. \]

In intermediate states, as soon as a new phase has arisen, or during the pressure drop of a two-phase system, subsequently each phase, as the pressure continues to fall, behaves in accordance with the general nature of this process. At some moment of the pressure drop the vapor phase begins to condense, and the liquid phase to evaporate. A further fall in pressure again acts on both the vapor and the liquid phase. The driving force of this process is the continuous fall in the pressure of the system and the tendency of the vapor and liquid phases to come into conformity with their thermodynamic state at the new, lower pressure. The phase ratio in the process of an elementary pressure drop of the system is

\[ (Y_0 - \Delta X) + (X_0 - \Delta Y) = 1. \]

The liquid phase will evaporate and decrease by the amount

\[ \Delta X = \frac{S_0' - S_1'}{r_1} T_{\mathrm{n}1} \cdot Y_0. \]

The vapor phase will condense and decrease by the amount

\[ \Delta Y = \frac{S_0'' - S_1''}{r_1} T_{\mathrm{n}1} \cdot X_0. \]

In the new state the system will have the form

\[ Y_1 + X_1 = 1. \]

From these co-

relation, it is evident that the formation of each phase is proportional to the amount of the initial phase, bearing in mind that \(S'_0-S'_1 \simeq S''_0-S''_1\); \(r_1\) and \(T_{\text{н}1}\) have the same value both for the vapor and for the liquid phase at pressure \(P_1\). From the equation \((Y'_0-\Delta X)+(X'_0-\Delta Y)=1\) and the dependence derived above, the conditions of the final directionality and the final result of adiabatic phase formation during isoentropic pressure release are determined: the quantitative transition and predominance of one process (AC, AE) over the other both in intermediate states and in the final state of the system’s pressure release.

For \(Y_0<\Delta X;\ \Delta Y>\Delta X\) — adiabatic condensation;

For \(Y_0>\Delta X;\ \Delta Y<\Delta X\) — adiabatic vaporization;

For \(Y_0=X_0;\ \Delta Y=\Delta X\) — AP — a process of equal phase formation.

The fundamental law of adiabatic phase formation and the quantitative characteristics of AP are well illustrated by the diagram of the phase state of \(\mathrm{H_2O}\) in \(T\)–\(S\) coordinates [1–3]. The phase-state diagram of \(\mathrm{H_2O}\) makes it possible to analyze the influence of the magnitude and interval of isoentropic pressure release, the initial and final pressures, and the ratio of the vapor and liquid phases on the phase-forming and energetic effect; to determine the limiting value of the new phase at the end of the process, the kinetic energy of the flow, etc., which are already known from the literature [1, 3].

The fundamental law of adiabatic phase formation, discovered on the basis of theoretical and experimental investigations, is, in turn, a consequence of the “mirror symmetry” of the processes of adiabatic phase formation, the identity of adiabatic vaporization and adiabatic condensation, and their being of one and the same nature, due to the identity and equality of the physical and thermodynamic properties and parameters for one and the same pressure level. For a given final pressure, the heat of vaporization during adiabatic condensation (index “c”) \(r_{\text{c}}\) is exactly equal to the heat of phase formation during adiabatic vaporization (index “v”) \(r_{\text{v}}\). The saturation temperatures of liquid and vapor do not differ, for the given final pressure, for the processes of AP and AC. The enthalpy of the liquid in these processes is \(i'_{\text{c}}=i'_{\text{v}}\), of saturated vapor \(i''_{\text{c}}=i''_{\text{v}}\), the entropies \(S'_{\text{c}}=S'_{\text{(v)}}\), \(S''_{\text{(c)}}=S''_{\text{(v)}}\). The surface tension of the liquid is one and the same both for the formation of droplets and for vapor bubbles for the given pressure level of the system. The same conditions are preserved for the nucleation of the vapor and liquid phases.

Another important property of adiabatic phase formation is the tendency of the system, in the process of pressure release, to reach the corresponding state appropriate to the given pressure level. At a given final pressure, a two-phase system under AP must necessarily have the physical and thermodynamic properties and parameters of saturated vapor and liquid.

The metastable state of the system and the disturbance of the equilibrium state under AP is a temporary, transient process that ensures nucleation of the liquid and vapor phases, a result of surface forces and of the bubble–droplet nature of the \(\mathrm{H_2O}\) liquid–vapor phase transitions. The quantitative and qualitative characteristics of adiabatic phase formation depend on the magnitude, interval, and character of the pressure release, and also on the ratio of the liquid and vapor phases in the initial state of the process. The conditions for nucleation of the vapor and liquid phases and the further development of the phases are identical both for AV and for AC under the same process conditions. At one and the same saturation temperature over a flat surface \(T_{\text{н}}\) and tabulated pressure \(P_{\text{п}}\), the pressu—

the vapor pressure in the bubble and its temperature under AP will be determined by the surface forces \(P_{\text{п}} = P_{\text{н}} - \dfrac{2\sigma}{\xi}\) and proves to be lower than \(T_{\text{н}}\). The pressure of the liquid in a droplet under these same conditions, on the contrary, proves to be higher by the same amount than the saturation pressure in the flow.

\[ P_{\text{ж}} = P_{\text{н}} + \frac{2\sigma}{\xi} \qquad P = P_{\text{п}} - P_{\text{н}} = P_{\text{ж}} - P_{\text{н}} = \pm \frac{2\sigma}{\xi}. \]

As a consequence of this, one and the same flow temperature simultaneously creates conditions for nucleation both for bubbles and for droplets of liquid in the flow. Each phase, for a given magnitude of the isoentropic pressure drop, in accordance with the fundamental law of phase formation, behaves in an entirely identical manner, arising and developing simultaneously with the same degree of intensity and in equal quantities per one and the same weight quantity of the liquid and vapor phases.

The interval of the pressure drop, in turn, will exert a different influence on the kinetics of adiabatic phase formation for one and the same magnitude of pressure drop [1, 3].

Adiabatic phase formation, as a process isolated from the external medium, proceeds in strict accordance with the material and energy balance of the system. The preservation of these balances determines the internal nature of the process of adiabatic phase formation, its specific properties and regularities. The change in the aggregate state of the system during pressure drop is the condition for compliance with the law of conservation of energy and matter for this process; it determines the phase-forming and energy effect, the identity of the AP and AC processes, and the kinetics of this process. During an isoentropic pressure drop of saturated liquid \((X_0 = 1)\) from \(P_0\) to \(P_1\), the available energy potential is determined by the enthalpy of the liquid (at \(C = 0\)) \(i'_0\) in the initial state, which cannot be changed at a lower pressure of the system. To satisfy this, the liquid is forced to change its aggregate state—to pass into a two-phase system with simultaneous generation of kinetic energy associated with the pressure drop.

\[ i'_0 = i'_1 \cdot X_1 + i''_1 \cdot Y_1 + H'_1; \qquad H'_1 = i'_0 - \left[i'_1 + (S'_0 - S'_1)T_{\text{н}1}\right] \]

During an isoentropic pressure drop of saturated vapor, the system is likewise forced to change its aggregate state in view of the different parameters of the liquid and vapor at the lower pressure and the transition of part of the thermal energy into the kinetic energy of the flow.

In the presence of irreversible processes, the energy efficiency (AF) is reduced and at the same time the phase-forming effect (AF) changes, without violation of the energy and material balance of the system. During an isoentropic pressure drop of a two-phase system, the change of each phase will also proceed in accordance with the fundamental law of AF.

\[ X_0 + Y_0 = X_1 + Y_1 = (X_0 - \Delta Y_1) + (Y_0 - \Delta X_1); \]

\[ \Delta Y_1 = \frac{S''_0 - S''_1}{r_1}\, T_{\text{н}1}\cdot X_0. \]

\[ \Delta X = \frac{S'_0 - S'_1}{r_1}T_{\text{н}1}\cdot Y_0; \qquad \frac{S''_0 - S''_1}{r_1}T_{\text{н}1} = \frac{S'_0 - S'_1}{r_1}T_{\text{н}1}, \]

\[ \Delta Y_1 = \alpha \cdot X_0; \qquad \Delta X_1 = \alpha \cdot Y_0. \]

The preservation of the material and energy balance and the action of the fundamental law of AF determine the limiting possibilities of this process.

in terms of kinetic energy and of the phase-forming effect. In particular, the system can never reach a single-phase state at the end of the process, as is possible in the process of a change in the state of aggregation with external heat exchange and at constant pressure.

The fundamental law of APF, the conservation of the internal energy and material balance of the system, determine the kinetics of adiabatic phase formation, and the phase-forming and energetic effects of this process. In this connection, the present-day concepts of the gas dynamics of two-phase media, of the kinetics of expansion of wet steam in steam turbines, the theoretical foundations of wet-steam turbines, and methods for experimental and theoretical investigations of moisture losses—constructed on the concepts of continuous ordinary growth of liquid droplets due to heat exchange, i.e., heat transfer from the liquid to the steam during its expansion in the flow section of wet-steam turbines—and other concepts [4, 5] not connected with the concept of adiabatic phase formation [4, 5], are unsuitable.

Such a traditional conception of the kinetics and physical essence of the expansion of wet steam in the nozzles of steam turbines has not even ensured the correct formulation of theoretical and experimental investigations of moisture losses in steam turbines in domestic and foreign practice [1, 6, 7].

LITERATURE

1. S. V. Polozhii. On the kinetics of the process of steam condensation in a turbine stage. Izv. TPI, vol. 137, 1965.

2. S. V. Polozhii. Steam-power installations with adiabatic steam formation. Izv. vuzov SSSR. Energetika, No. 1, 1965.

3. S. V. Polozhii. Adiabatic phase formation. Information bulletin on the problem of converting thermal power plants to adiabatic phase formation. TPI, No. 4, 1969.

4. M. E. Deich, G. A. Filippov. Gas Dynamics of Two-Phase Media. Energiya, 1968.

5. I. I. Kirillov, R. M. Yablonik. Fundamentals of the Theory of Wet-Steam Turbines. Mashinostroenie, 1968.

6. S. V. Polozhii. On the question of the influence of steam moisture on the efficiency of a turbine stage. Izv. vuzov SSSR. Energetika, No. 7, 1962.

7. S. V. Polozhii. Once again on the question of the influence of steam moisture on the efficiency of a turbine stage. Izv. vuzov SSSR. Energetika, No. 3, 1963.